Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-11T02:35:16.819Z Has data issue: false hasContentIssue false
  • English
  • Français

UNFOLDING THE DOUBLE SHUFFLE STRUCTURE OF $q$-MULTIPLE ZETA VALUES

Part of: Basic hypergeometric functions Zeta and $L$-functions: analytic theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  24 April 2015

JAIME CASTILLO-MEDINA ,
KURUSCH EBRAHIMI-FARD  and
DOMINIQUE MANCHON
JAIME CASTILLO-MEDINA
Affiliation:
Universidad de València, Facultat de Ciències Matemàtiques, C/-Doctor Moliner, 50, 46100 Burjassot-Valencia, Spain email jaicasme@alumni.uv.es
KURUSCH EBRAHIMI-FARD
Affiliation:
Instituto de Ciencias Matemáticas, C/-Nicolás Cabrera, no. 13–15, 28049 Madrid, Spain email kurusch@icmat.es, kurusch.ebrahimi-fard@uha.fr
DOMINIQUE MANCHON*
Affiliation:
Université Blaise Pascal, CNRS–UMR 6620, 63177 Aubière, France email manchon@math.univ-bpclermont.fr
*
✉manchon@math.univ-bpclermont.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We exhibit the double $q$-shuffle structure for the $q$MZVs introduced by Ohno et al. [‘Cyclic $q$-MZSV sum’, J. Number Theory132 (2012), 144–155].

Keywords

multiple zeta valuesbasic hypergeometric functionsq-analogueshuffle relationdouble shuffle

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11G55: Polylogarithms and relations with $K$-theory 33D70: Other basic hypergeometric functions and integrals in several variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 368 - 388
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Ahlfors, L. V., Complex Analysis, 3rd edn (McGraw-Hill Higher Education, New York, 1980).Google Scholar
Bachmann, H. and Kühn, U., ‘The algebra of multiple divisor functions and applications to multiple zeta values’, arXiv:1309.3920.Google Scholar
Bailey, W. N., ‘An algebraic identity’, J. Lond. Math. Soc. 11 (1936), 156–160.Google Scholar
Bradley, D. M., ‘Multiple q-zeta values’, J. Algebra 283 (2005), 752–798.CrossRefGoogle Scholar
Cartier, P., ‘Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents’, Sémin. Bourbaki 885 (2001), 137–173.Google Scholar
Castillo Medina, J., Ebrahimi-Fard, K. and Manchon, D., ‘On Euler’s decomposition formula for q-MZVs’, Ramanujan J. to appear; doi:10.1007/s11139-014-9638-8.Google Scholar
Castillo Medina, J., Ebrahimi-Fard, K. and Manchon, D., ‘Unfolding the double shuffle structure of $q$-MZVs’, arXiv:1310.1330.Google Scholar
Cherednik, I., ‘On q-analogues of Riemann’s zeta function’, Selecta Math. (N.S.) 7(4) (2001), 447–491.CrossRefGoogle Scholar
Díaz, R. and Páez, M., ‘An identity in Rota–Baxter algebras’, Sém. Lothar. Combin. 57 (2007), B57b, 8 pages.Google Scholar
Ebrahimi-Fard, K. and Patras, F., ‘La structure combinatoire du calcul intégral’, Gaz. Math. 138 (2013), 5–22. Société Mathématique de France (SMF). English translation: ‘Rota–Baxter algebra. The combinatorial structure of integral calculus’, arXiv:1304.1204v1.Google Scholar
Guo, L. and Keigher, W., ‘On differential Rota–Baxter algebras’, J. Pure Appl. Algebra 212 (2008), 522–540.Google Scholar
Hoffman, M. E., ‘The algebra of multiple harmonic series’, J. Algebra 194(2) (1997), 477–495.Google Scholar
Hoffman, M. E., ‘Quasi-shuffle products’, J. Algebraic Combin. 11 (2000), 49–68.Google Scholar
Hoffman, M. E., ‘Algebraic aspects of multiple zeta values’, in: Zeta Functions, Topology and Quantum Physics (eds. Aoki, T. et al. ) (Springer, New York, 2005), 51–74.CrossRefGoogle Scholar
Ihara, K., Kaneko, M. and Zagier, D., ‘Derivation and double shuffle relations for multiple zeta values’, Compositio Math. 142 (2006), 307–338.Google Scholar
Kaneko, M., Kurokawa, N. and Wakayama, M., ‘A variation of Euler’s approach to values of the Riemann zeta function’, Kyushu J. Math. 57 (2003), 175–192.Google Scholar
Ohno, Y., Okuda, J. and Zudilin, W., ‘Cyclic q-MZSV sum’, J. Number Theory 132 (2012), 144–155.Google Scholar
Okounkov, A., ‘Hilbert schemes and multiple q-zeta values’, Funct. Anal. Appl. 48(2) (2014), 138–144.CrossRefGoogle Scholar
Racinet, G., ‘Doubles mélanges des polylogarithmes multiples aux racines de l’unité’, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 185–231.Google Scholar
Rota, G.-C., ‘Baxter operators, an introduction’, in: Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Contemporary Mathematicians (ed. Kung, J. P. S.) (Birkhäuser, Boston, MA, 1995).Google Scholar
Rota, G.-C. and Smith, D. A., ‘Fluctuation theory and Baxter algebras’, Symp. Math. IX (1971), 179–201.Google Scholar
Schlesinger, K. G., ‘Some remarks on $q$-deformed multiple polylogarithms’, Preprint, arXiv:math/0111022, 2001.Google Scholar
Takeyama, Y., ‘The algebra of a q-analogue of multiple harmonic series’, Symmetry Integrability Geom. Methods Appl. 9 (2013), 061.Google Scholar
Waldschmidt, M., ‘Valeurs, zêta multiples: une introduction’, J. Théor. Nombres Bordeaux 12 (2000), 581–595.Google Scholar
Waldschmidt, M., ‘Multiple polylogarithms’, Lecture at Institute of Mathematical Sciences, Chennai, November 2000, available at http://www.math.jussieu.fr/ miw/articles/.Google Scholar
Zagier, D., ‘Values of zeta functions and their applications’, in: First European Congress of Mathematics, Progress in Mathematics, 120, Vol. II (Birkhäuser, Basel, 1994), 497–512.Google Scholar
Zhao, J., ‘q-Multiple zeta functions and q-multiple polylogarithms’, Ramanujan J. 14 (2007), 189–221.CrossRefGoogle Scholar
Zudilin, W., ‘Algebraic relations for multiple zeta values’, Russian Math. Surveys 58(1) (2003), 1–29.Google Scholar